Denote by $\mathcal{G}(n,c,g,k)$ the set of all connected graphs order $n$, having $c$ cycles, girth $g$, and $k$ pendant vertices. In this paper, we give a partial characterisation structure those in maximising number induced subgraphs. For special case where $c=1$, find complete maximal unicyclic graphs. We also derive precise formula for corresponding maximum given following parameters: (1) ...

In this paper, the signed graphs with one positive eigenvalue are characterized, and the signed graphs with pendant vertices having exactly two positive eigenvalues are determined. As a consequence, the signed trees, the signed unicyclic graphs and the signed bicyclic graphs having one or two positive eigenvalues are characterized.

The principal ratio of a connected graph, denoted γ(G), is the ratio of the maximum and minimum entries of its first eigenvector. Cioabă and Gregory conjectured that the graph on n vertices maximizing γ(G) is a kite graph: a complete graph with a pendant path. In this paper we prove their conjecture.

In this paper the possible numbers of blocks lEI n E21 in common to two G-designs, (ti, Er) and (V, B 2 ), are determined, where the graph G has six vertices and six edges, contains a cycle of length four, and has two pendant edges. There are four such graphs G.

The bull is a graph consisting of a triangle and two pendant edges. A graphs is called bull-free if no induced subgraph of it is a bull. In this paper we prove that every bull-free graph on n vertices contains either a clique or a stable set of size n 1 4 , thus settling the Erdős-Hajnal conjecture [5] for the bull.

A bull is a graph obtained by adding a pendant vertex at two vertices of a triangle. Here we present polynomial-time combinatorial algorithms for the optimal weighted coloring and weighted clique problems in bull-free perfect graphs. The algorithms are based on a structural analysis and decomposition of bull-free perfect graphs.

In this paper, the trees with the largest Dirichlet spectral radius among all trees with a given degree sequence are characterized. Moreover, the extremal graphs having the largest Dirichlet spectral radius are obtained in the set of all trees of order n with a given number of pendant vertices.